In this paper we use the theory of generalized geometric programming to develop a dual for a discrete time convex optimal control problem. This has interesting interpretational implications. Further it is shown that the variables in the dual problem are intimately related to the costate vector in the usual Maximum Principle approach.

The dispersion of a passive solute in three-dimensional flow is examined for short times after the injection of solute. If the diffusivity is constant, the solute at first diffuses isotropically about the fluid particle originally coincident with the injection point whilst, at longer times, the effect of diffusion across a velocity shear becomes more important. An asymptotic expansion is derived for the concentration of solute at small times after its injection into the fluid flow and the use of the theory is illustrated for three representative flows. Some critical remarks on the applicability and limitations of the results conclude the note.

Consideration of functions whose second difference along the trajectories of a difference equation is positive gives a stability theorem for autonomous discrete-time systems. Such functions can be used to estimate domains of nonglobal stability.

Maximum and minimum principles for capillary surface problems with prescribed contact angle are derived in a unified manner from canonical variational theory. The results are illustrated by calculations for a liquid in a cylindrical container with circular cross-section.

Tensor identities for finite dimensional representations of arbitrary semi-simple Lie algebras are derived and are applied to the construction of left-projection operators which project out the shift components of tensor operators from the left. The corresponding adjoint identities are also derived and are used for the construction of right-projection operators. It is also shown that, on a finite dimensional irreducible representation, these identities may be considerably reduced. Commutation relations between the shift tensors of a tensor operator are also computed in terms of the roots appearing in the tensor identities.

In this paper, a computational algorithm for solving a class of optimal control problems involving discrete time-delayed arguments is presented. By way of example, a simple model of a production firm is devised for which the algorithm is used to solve a decision-making problem.

Einstein's vacuum field equations of an axially symmetric stationary rotating source are studied. Using the oblate spheroidal coordinate system, a class of asymptotically fiat solutions representing the exterior gravitational field of a stationary rotating oblate spheroidal source is obtained. Also it is proved that an analytic axisymmetric and stationary distribution of dust cannot be the source for the gravitational field described by the axisymmetric stationary metric.

It has been known for some time that if a certain non-degeneracy condition is satisfied then the successive solution estimates x(r) produced by barrier function techniques lie on a smooth trajectory. Accordingly, extrapolation methods can be used to calculate x(0). In this paper we analyse the situation further treating the special case of the log barrier function. If the non-degeneracy assumption is not satisfied then the approach to x(0) is like r½ rather than like r which would be expected in the non-degenerate case. A measure of sensitivity is introduced which becomes large when the non- degeneracy assumption is close to violation, and it is shown that this sensitivity measure is related to the growth of dix/dri with respect to i for fixed r small enough on the solution trajectory. With this information it is possible to analyse the extrapolation procedure and to predict the number of stages of extrapolation which are useful.

Sufficient conditions for an equilibrium point to be an attractor or a global attractor are derived for a class of first-order difference equations which need not be continuous at the equilibrium point. These conditions involve Lyapunovlike functions which need not be continuous and are applied to the logistic equation with a piecewise continuous control.

A one-sided one-dimensional random walk with repulsion from the origin is solved exactly. The walk imitates the self-avoiding walk problem insofar as the mean end-to-end distance of an n-step walk tends asymptotically to n as n tends to infinity.